Groups of volume-preserving diffeomorphisms of noncompact manifolds and mass flow toward ends

TL;DR

This paper extends Moser's theorem to noncompact manifolds, demonstrating how volume-preserving diffeomorphisms can realize mass transfer toward ends, and establishes a continuous section for the end charge homomorphism.

Contribution

It introduces a unified approach for mass transfer realization and extends Moser's theorem to noncompact manifolds, providing new deformation retraction results for diffeomorphism groups.

Findings

Extension of Moser's theorem to noncompact manifolds.

Existence of a continuous section for the end charge homomorphism.

Deformation retraction of certain diffeomorphism subgroups.

Abstract

Suppose M is a noncompact connected oriented C^infty n-manifold and omega is a positive volume form on M. Let D^+(M) denote the group of orientation preserving diffeomorphisms of M endowed with the compact-open C^infty topology and D(M; omega) denote the subgroup of omega-preserving diffeomorphisms of M. In this paper we propose a unified approach for realization of mass transfer toward ends by diffeomorphisms of M. This argument together with Moser's theorem enables us to deduce two selection theorems for the groups D^+(M) and D(M; omega). The first one is the extension of Moser's theorem to noncompact manifolds, that is, the existence of sections for the orbit maps under the action of D^+(M) on the space of volume forms. This implies that D(M; omega) is a strong deformation retract of the group D^+(M; E^omega_M) consisting of h in D^+(M) which preserves the set E^omega_M of omega-finite ends of M. The second one is related to the mass flow toward ends under volume-preserving diffeomorphisms of M. Let D_{E_M}(M; omega) denote the subgroup consisting of all h in D(M; omega) which fix the ends E_M of M. S.R.Alpern and V.S.Prasad introduced the topological vector space S(M; omega) of end charges of M and the end charge homomorphism c^omega : D_{E_M}(M; omega) to S(M; omega), which measures the mass flow toward ends induced by each h in D_{E_M}(M; omega). We show that the homomorphism c^omega has a continuous section. This induces the factorization D_{E_M}(M; omega) cong ker c^omega times S(M; omega) and it implies that ker c^omega is a strong deformation retract of D_{E_M}(M; omega).